Mathardy! (G7/SHSAT Prep)

Percents and Algebra

100

A shirt that originally costs $80 goes on sale for a price of $45. What is the percent discount?

Percent change = ((new-old)/old) x 100

Percent change = (45-80)/80*100 

Percent change = (35/80)*100 

ANS: 43.75%

200

Find the GCF and LCM of the following numbers: 24, 30, and 48.

24 = 2 x 2 x 2 x 3

30 = 2 x 3 x 5

48 = 2 x 2 x 2 x 2 x 3

GCF = 2 x 3 = 6

LCM = 2 x 2 x 2 x 2 x 3 x 5 = 240 

ANS: GCF = 6, LCM = 240

300

What is the GCF of 24x⁷y⁴ and 9x⁵y⁷ ?

GCF =  3x^5y^4 

400

38% of a number is equal to 24. What is the number rounded to the hundredths place?

0.38x=24

x=24/0.38

x=63.157

x=63.16

500

Simplify: 

5(1+0.3)^2-5

5(1.3)^2-5

5(1.69)-5

8.45-5

3.45

Geometry and Radicals

100

Simplify:

sqrt88

Prime factorization of 88 = 2 x 2 x 2 x 11

= 2² x 2 x 11

=sqrt(2^2)*sqrt2*sqrt11

OR

sqrt88=sqrt4*sqrt22

sqrt88=2sqrt22

200

The diameter of a circle is 8 inches. What is the area of the circle in square inches? Leave your answer in terms of pi.

A = 

pir^2=pi(4)^2=16pi

300

A triangle has three interior angles. The largest angle is 3 times the smallest angle. The 2nd largest angle is double the smallest angle. What is the measure of the largest angle?

Let x = the smallest angle

2nd largest angle = 2x

Largest angle = 3x 

Total degrees in a triangle = 180 = x + 2x + 3x 

180 = 6x 

x = 30 

Largest angle = 3x = 3(30) = 90

ANS: 90

400

Find the area of this trapezoid: 

Area = 

(1/2)(base 1 + base 2)(height)

(1/2)(12+50)(28)

868 

500

The measure of 1 interior angle of a regular polygon is 162 degrees. How many sides does the polygon have?

((n-2)*180)/n=162

(n-2)*180=162n

180n-360=162n

-360=-18n

n=20

Exponents and Scientific Notation

100

Express the following number in scientific notation: 0.0000234

2.34 x 10^-5

200

Simplify the expression: (3.15 x 10^-3) x (2.4 x 10⁵)

3.15 x 2.4 = 7.56 

10^-3 x 10⁵ = 10²

ANS: 7.56 x 10²

300

Simplify: 

(8x^3y^8)/((2xy^2)^3

 Your answer must not have negative exponents

(8x^3y^8)/((2xy^2)^3

(8x^3y^8)/(8x^3y^6)

y^2

400

Simplify the product of (3x³y^-2)² with (2x²y⁵)

(3x³y^-2)² (2x²y⁵) 

(9x⁶y^-4)(2x²y⁵)

9*2 = 18 

x⁶*x²=x⁸

y^-4*y⁵=y

ANS: 18x⁸y¹

500

Solve for x if: 3²⁷ = 9^3x

3²⁷ = 9^3x

9 = 3²

9^3x = (3²)^3x

(3²)^3x = 3^6x

3²⁷ = 3^6x

27 = 6x 

x = 27/6 = 9/2

ANS: 9/2

Statistics and Probability

100

A quarter has two sides: Heads and Tails. If you flip a quarter two times, what is the probability of getting Heads twice?

P(heads first time) x P(heads second time)

(1/2)(1/2) = 1/4

ANS: 1/4

200

You flip a coin 5 times and it lands on Heads 5 times. What is the probability that the next flip (the 6th flip) will land on Heads as well?

ANS: 1/2

300

A student is about to take a final chemistry test. She currently has a class average of 80 across her first 4 tests. The final is going to be worth 2 test grades. If she wants to get an average of 85 in the class, what must she get on the final exam?

Average = (sum of all test scores / # of tests)

Sum of all test scores = average x # of tests

Current sum after 4 tests = 4 x 80 = 320 points 

Desired sum = 6 x 85 = 510 points

510 - 320 = 190 points 

190/2 = 95 on the final

ANS: 95

 

400

You roll 2 standard 6 - sided dice. What is the probability that the sum of the two dice will be equal to 6? 

Total combinations = 6 x 6 = 36

Total ways to getting 6: (1,5) (5,1) (2,4)(4,2)(3,3)

ANS: 5/36                                     

500

What is the mean, median and mode for the following data set?

25, 22, 17, 10, 11, 12, 15, 16, 25

Data set:

10, 11, 12, 15, 16, 17, 22, 25, 25

Mean = (sum of all values / # of values)

Mean = (10+11+12+15+16+17+22+25+25) /  9

Mean = 17

Median = value in the middle when arranged in order

Median = 16

Mode = 25

Ratios and Proportions

100

3 workers can paint 5 houses in a week. How many houses can 12 workers paint in a week?

3 workers: 5 houses: 1 week

12 workers: x houses: 1 week

5 x 4 = 20 houses

ANS: 20

200

There are two triangles that are similar:

Triangle ABC is similar to triangle XYZ. The length of AB = 5, and length of the BC is 6. If the length of XY is 8, what is the length of YZ?

 (AB)/(BC)=(XY)/(YZ) 

 5/6=8/(YZ) 

5(YZ)=8(6)

5YZ = 48

YZ=

48/5

300

The equation,

y=4/5x

describes a proportional relationship and forms a straight line on the graph. A point with a y-coordinate of 5 lies on the line. What is the corresponding x-coordinate?

 y=(4/5)(x) 

5=(4/5)(x)

x = 5(5/4)

x = (25/4)

400

In a jar, the ratio of quarters to dimes to nickels is 6:8:5 respectively. If the total value of the coins in the jar is $7.65, how many quarters, dimes and nickels are there?

6(0.25) + 8(0.1) + 5(0.05) = $2.55

$7.65/$2.55 = 3

3 x (6:8:5) = 18:24:15

quarters = 18

dimes = 24

nickels = 15

500

There are two similar rectangles where each corresponding side of the large rectangle is triple that of the smaller rectangle.  What would be the ratio of the area of the smaller rectangle to that of the larger rectangle?

Area of smaller rectangle = length x width

Area of larger rectangle = 3length x 3 width = 9 lw

The ratio of the small to large rectangle = 1:9

ANS: 1:9

Extra Hard (Each 500 points)

100

You are building a rectangular fence and have 80 feet of fencing. You are going to use all 80 feet of fencing. What is the maximum possible area for the fence?

Max area of a rectangle is the shape of a square 

80 feet is the perimeter each side will be 20 feet

Max area will be 20² = 400 feet²

200

A squirrel can run at a speed of 120 feet per min. How fast is that in miles per day? Answer must be a simplified fraction

1 mile = 5280 feet

(120 feet)/(1 min)*(60 min)/(1 hour)*(24hours)/(1day)*(1 mi)/(5280)

ANS: 

360/11 or 32 8/11

300

FOIL and Simplify : 

(x+2)(x-3)(x-2)

(x+2)(x-3)(x-2)

(x^2-3x+2x-6)(x-2)

(x^2-x-6)(x-2)

(x^3-2x^2-x^2+2x-6x+12)

x^3-3x^2-4x+12

400

There are 6 different cookies in a tray. You get to choose 3 of them. How many different combinations of cookies can you take? 

Let cookies be A,B,C,D,E,F

Combinations: ABC, ABD, ABE, ABF

                      ACD, ACE, ACF

                      ADE, ADF

                      AEF

                      BCD, BCE, BCF, etc. 

Faster method: Find how much combinations can be made with the same three cookies. 

ABC = ACB = BAC = BCA = CAB = CBA 

Combos = ((6)(5)(4))/6=20 

ANS: 20 

500

The equation for the volume of a sphere is 

4/3pir^3

How many times does the volume increase if a sphere's radius is quadrupled? 

V=4/3pir^3

4/3pi(4r)^3

 Only difference is r^3 vs (4r)^3

(4r)^3=64r^3

 Volume will be 64 times larger